Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,\Delta)$ revisited

TL;DR

This paper extends previous results on matrix factorizations to odd-dimensional orthogonal and unitary groups, providing new proofs of their classification theorems under certain conditions.

Contribution

It generalizes matrix decomposition techniques to $O_{2n+1}(R)$ and $U_{2n+1}(R, riangle)$, leading to simplified proofs of their Sandwich Classification Theorems.

Findings

Established matrix expression results for $O_{2n+1}(R)$ and $U_{2n+1}(R, riangle)$

Provided shorter proofs of the Sandwich Classification Theorems for these groups

Extended previous matrix factorization techniques to new algebraic groups

Abstract

In a recent paper, the author proved that if $n\geq 3$ is a natural number, $R$ a commutative ring and $\sigma\in GL_n(R)$, then $t_{kl}(\sigma_{ij})$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ matrices of the form $^{\epsilon}\sigma^{\pm 1}$ where $\epsilon\in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,\Delta)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and $U_{2n+1}(R,\Delta)$.